基泽曼幺半群的基数及其奇偶渐近行为
Cardinalities of Kiselman monoids and their parity asymptotics
AI总结：
本文证明基泽曼半群$K_n$的奇偶渐近极限$c_{\mathrm{even}}$与$c_{\mathrm{odd}}$不相等，引入基泽曼变换构造，通过线性递推扩展OEIS序列A125625的$K_n$值并给出极限差的严格数值界。
摘要：
已知基泽曼半群$K_n$的基数随$n$呈双指数增长：序列$2^{-n/2}\log_2|K_n|$沿偶数和奇数$n$子序列收敛到正极限$c_{\mathrm{even}}$和$c_{\mathrm{odd}}$，此前人们认为这两个极限相等。本文证明它们实际上是不同的。由于基泽曼半群是$\mathcal{L}$-平凡的，后缀关系在$K_n$上定义了一个偏序。我们引入偏序集$P$的\emph{基泽曼变换}$\mathsf{Kis}(P)$，这是一种从$K_n$对应的偏序集恢复$K_{n+1}$对应偏序集的构造。对$P$和$\mathsf{Kis}(P)$中的严格递增链计数，可得到由具有显式二项式项的对称无限矩阵控制的线性递推关系，该关系可用于高效计算$|K_n$，使OEIS序列A125625中列出的值能扩展到$n=14$之外。生成函数分析给出了上述渐近极限的公式，以及严格数值界，确定$c_{\mathrm{even}} - c_{\mathrm{odd}} = 1.9945066605\ldots\times10^{-6}$。
英文摘要：
The cardinalities of Kiselman's semigroups $K_n$ are known to grow double exponentially in $n$: the sequence $2^{-n/2}\log_2|K_n|$ converges along the even and odd $n$ subsequences to positive limits $c_{\mathrm{even}}$ and $c_{\mathrm{odd}}$ that appear to coincide. In this paper we prove that they are indeed distinct. Since Kiselman's semigroups are $\mathcal{L}$-trivial, the suffix relation defines a partial order on $K_n$. We introduce the \emph{Kiselman transform} $\mathsf{Kis}(P)$ of a poset $P$, a construction recovering the poset associated with $K_{n+1}$ from that associated with $K_n$. Counting strictly increasing chains in $P$ and $\mathsf{Kis}(P)$ yields a linear recurrence governed by a symmetric infinite matrix with explicit binomial entries. This can be used towards efficiently computing $|K_n|$, allowing the values listed in OEIS sequence A125625 to be extended beyond $n=14$. A generating functions analysis gives formulas for the above asymptotic limits, together with rigorous numerical bounds establishing that \[ c_{\mathrm{even}} - c_{\mathrm{odd}} = 1.9945066605\ldots\times10^{-6} \]
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
